69
3.2. The Dirac equation
(with obvious cyclic permutations), which are precisely the commutation relations satisfied by the components J ˆ x , J ˆ y and J ˆ z of the angular momentum
operator J ˆ in quantum mechanics (see appendix A). Furthermore, the eigen1
values of S z are ±
1 , and of S
2 are s(s + 1) with s = . So these matrices
2
2
undoubtedly represent quantum mechanical angular momentum operators,
1
appropriate to a state with angular momentum quantum number j = . This
2
is precisely what ‘spin’ is. We will discuss this in more detail in section 3.3.
It is important to note that the choice (3.31) of α and β is not unique. In
fact, all matrices related to these by any unitary 4 × 4 matrix U (which thus
preserves the anticommutation relations) are allowed:
α
′
i
= Uα i U
−1
(3.38)
β
′
= UβU
−1 .
(3.39)
Another commonly used representation is provided by the matrices
α =
( σ
0
0
− σ
)
β =
(
0 1
1 0
)
.
(3.40)
The reader may check (problem 3.2) that these matrices also satisfy (3.34)–
(3.36).
Unless otherwise stated, we shall use the standard representation (3.31).
This is generally convenient for ‘low energy’ applications – that is, when the
momentum |p| is significantly smaller than the mass m. In that case, βm will
be the largest term in the Dirac Hamiltonian (see (3.23)), and it is sensible
to have it in diagonal form. The choice (3.40), by contrast, is more natural
when the mass is small compared with the energy or momentum.
3.2.1 Free-particle solutions
Since the Dirac Hamiltonian now involves 4 × 4 matrices, it is clear that we
must interpret the Dirac wavefunction ψ as a four-component column vector –
the so-called Dirac spinor. Let us look at the explicit form of the free-particle
solutions. As in the KG case, we look for solutions in which the space–time
behaviour is of plane-wave form and put
−ip·x
ψ = ωe
(3.41)
μ
where ω is a four-component spinor independent of x, and e
−ip·x , with p =
(E, p), is the plane-wave solution corresponding to 4-momentum p
μ . We substitute this into the Dirac equation
i∂ψ/∂t = (−iα · ∇ + βm)ψ
(3.42)
using the explicit α and β matrices. In order to use the 2 × 2 block form, it is
conventional (and convenient) to split the spinor ω into two two-component
spinors φ and χ:
( )
φ
ω =
.
(3.43)
χ
3.2. The Dirac equation
(with obvious cyclic permutations), which are precisely the commutation relations satisfied by the components J ˆ x , J ˆ y and J ˆ z of the angular momentum
operator J ˆ in quantum mechanics (see appendix A). Furthermore, the eigen1
values of S z are ±
1 , and of S
2 are s(s + 1) with s = . So these matrices
2
2
undoubtedly represent quantum mechanical angular momentum operators,
1
appropriate to a state with angular momentum quantum number j = . This
2
is precisely what ‘spin’ is. We will discuss this in more detail in section 3.3.
It is important to note that the choice (3.31) of α and β is not unique. In
fact, all matrices related to these by any unitary 4 × 4 matrix U (which thus
preserves the anticommutation relations) are allowed:
α
′
i
= Uα i U
−1
(3.38)
β
′
= UβU
−1 .
(3.39)
Another commonly used representation is provided by the matrices
α =
( σ
0
0
− σ
)
β =
(
0 1
1 0
)
.
(3.40)
The reader may check (problem 3.2) that these matrices also satisfy (3.34)–
(3.36).
Unless otherwise stated, we shall use the standard representation (3.31).
This is generally convenient for ‘low energy’ applications – that is, when the
momentum |p| is significantly smaller than the mass m. In that case, βm will
be the largest term in the Dirac Hamiltonian (see (3.23)), and it is sensible
to have it in diagonal form. The choice (3.40), by contrast, is more natural
when the mass is small compared with the energy or momentum.
3.2.1 Free-particle solutions
Since the Dirac Hamiltonian now involves 4 × 4 matrices, it is clear that we
must interpret the Dirac wavefunction ψ as a four-component column vector –
the so-called Dirac spinor. Let us look at the explicit form of the free-particle
solutions. As in the KG case, we look for solutions in which the space–time
behaviour is of plane-wave form and put
−ip·x
ψ = ωe
(3.41)
μ
where ω is a four-component spinor independent of x, and e
−ip·x , with p =
(E, p), is the plane-wave solution corresponding to 4-momentum p
μ . We substitute this into the Dirac equation
i∂ψ/∂t = (−iα · ∇ + βm)ψ
(3.42)
using the explicit α and β matrices. In order to use the 2 × 2 block form, it is
conventional (and convenient) to split the spinor ω into two two-component
spinors φ and χ:
( )
φ
ω =
.
(3.43)
χ